Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 203 / 4 / c

Write $h=h_U+h^{\mathbb{D}\setminus U}$ with $U=B(0,e^{-t})$. By part (b), the harmonic part contributes the same value $h^{\mathbb{D}\setminus U}(0)$ to every inner circle average. At radius $e^{-t}$ the zero-boundary part contributes zero; equivalently, take the limit from inner circles and use the assumed continuity. HenceThe field $h_U$ is independent of $h^{\mathbb{D}\setminus U}$, so the increment has the required independence.

The dilation $z\mapsto e^{t}z$ maps $U$ onto the unit disc and maps the circle of radius $e^{-s}$ onto the circle of radius $e^{-(s-t)}$. The Conformal invariance of the two-dimensional Gaussian free field therefore gives